This disclosure is related to the field of electromagnetic induction well logging. More specifically, the disclosure relates to methods for processing multiaxial electromagnetic induction measurements to obtain orientation of electrical anisotropy and formation layer boundaries.
FIG. 2 schematically illustrates such a tri-axial tool 10 and the component tensor measurement C. The instrument 10 may include one or more multi-axial electromagnetic transmitters T disposed on the instrument 10, and have one or more multi-axial electromagnetic receivers (each receiver usually consisting of a main receiver RM and a balancing or “bucking” receiver RB to attenuate direct induction effects) at one or more axially spaced apart positions along the longitudinal axis z of the tool 10. An instrument sold under the trademark RT SCANNER, which is a trademark of Schlumberger Technology Corporation, Sugar Land, Tex., uses triaxial transmitters and receivers, wherein the transmitters and receivers have three, mutually orthogonal coils having magnetic dipole axes oriented along the tool axis z and along two other mutually orthogonal directions shown at x and y. The instrument measurements in the present example may be obtained in the frequency domain by energizing the transmitter T with a continuous wave (CW) alternating current having one or more discrete frequencies (using more than one discrete frequency may enhance the signal-to-noise ratio). However, measurements of the same information content may also be obtained using time domain signals through a Fourier decomposition process by energizing the transmitter T with one or more types of transient currents. This is a well-known physics principle of frequency-time duality. Voltages induced in each coil of one of the receivers RM/RB is shown in the tensor C represented by the voltage V with a two letter subscript as explained above representing the axis (x, y or z) of the transmitter coil used and the axis of the receiver coil (x, y or z) used to make the particular voltage measurements. The voltage measurements in tensor C may be processed to obtain the described apparent conductivity tensors. Subsurface formation properties, such as horizontal and vertical conductivities (σh, σv) or their inverse, horizontal and vertical resistivities (Rh, Rv), relative dip angle (θ) and the dip azimuthal direction (ϕ), as well as borehole/tool properties, such as drilling fluid (mud) conductivity (amud), wellbore diameter (hd), tool eccentering distance (decc), tool eccentering azimuthal angle (ψ), all affect the measurements of voltages used to determine the conductivity tensors. Multi-axial induction well logging instruments known in the art may include one or more transmitters of the type shown in FIG. 2 and a plurality of receivers such as shown in FIG. 2, each at a different longitudinal spacing (i.e., along the z axis or longitudinal axis of the instrument).
A corresponding multi-axial electromagnetic measurement system may be implemented in a logging while drilling (LWD) instrument. In LWD instruments, the receivers may be implemented by spaced apart tilted or transverse coil sets disposed on the exterior of a drill collar. Rather than measuring amplitude and phase of induced voltages in the receiver, an amplitude reduction or ratio of the detected voltage in the spaced apart coil sets may be measured, as well as a phase shift in the detected voltage between the receiver coil sets. An example LWD instrument is sold under the trademark PERISCOPE, which is also a trademark of Schlumberger Technology Corporation.
Measurements known in the art for determining direction of and distance to boundaries between formation layers having different electrical conductivities, and/or resistivity anisotropy and its orientation are based on the property of XZ−ZX and XZ+ZX couplings (symmetrized and antisymmetrized directional voltage measurements) that enhance or reduce sensitivity to bed boundaries or formation anisotropy and dip [see references 1,2 and 3]. These properties (removal or amplification of dip and anisotropy effect) are exact in transversely isotropic (TI) formations for electromagnetic induction type measurements, as the one implemented in the wireline RTSCANNER instrument, and are approximate for electromagnetic propagation measurements, such as implemented in the PERISCOPE LWD instrument. Example layouts for electromagnetic induction and propagation type measurements are described in certain U.S. patents [see references 4, and 5].
The raw PERISCOPE instrument directional measurements (voltages) may be continuously acquired while the instrument is rotating, e.g., to advance or drill a wellbore through subsurface formations. The fitting algorithm [see references 1 and 2] outputs Fourier coefficients for each frequency, f, transmitter, t, and receiver r:
                              V          ⁡                      (                          f              ,              t              ,              r                        )                          =                              a            0                    +                                    ∑                              k                =                1                            2                        ⁢                          {                                                                    a                    k                                    ⁢                                                                          ⁢                                      cos                    ⁡                                          (                                              k                        ⁢                                                                                                  ⁢                        ϕ                                            )                                                                      +                                                      b                    k                                    ⁢                                      sin                    ⁡                                          (                                              k                        ⁢                                                                                                  ⁢                        ϕ                                            )                                                                                  }                                                          (        1        )            where ak and bk are complex quantities, the subscript k denotes the number of the harmonic of the base frequency f, and ϕ defines the bedding (formation layer boundary or anisotropy) azimuth. Consider only the z-transverse (where z is oriented along the tool axis) coupling for the 1st harmonic measurement:Vz-transverse=a1 cos ϕ+b1 sin ϕ  (2)
The original boundary orientation angle with respect to the tool axis is determined by weighted averaging of angles for the real and imaginary part of the detected voltages, to minimize the noise effect assuming that angles from the real and imaginary part of voltage are the same [see reference 1]:
                    ϕ        =                                                                                                                                                                          Re                          ⁢                                                                                    {                                                              b                                1                                                            }                                                        2                                                                          +                                                  Re                          ⁢                                                                                    {                                                              a                                1                                                            }                                                        2                                                                                                                ⁢                                          tan                                              -                        1                                                              ⁢                                                                  Re                        ⁢                                                  {                                                      b                            1                                                    }                                                                                            Re                        ⁢                                                  {                                                      a                            1                                                    }                                                                                                      +                                                                                                                                                                        Im                        ⁢                                                                              {                                                          b                              1                                                        }                                                    2                                                                    +                                              Im                        ⁢                                                                              {                                                          a                              1                                                        }                                                    2                                                                                                      ⁢                                      tan                                          -                      1                                                        ⁢                                                            Im                      ⁢                                              {                                                  b                          1                                                }                                                                                    Im                      ⁢                                              {                                                  a                          1                                                }                                                                                                                                                                                      Re                  ⁢                                                            {                                              b                        1                                            }                                        2                                                  +                                  Re                  ⁢                                                            {                                              a                        1                                            }                                        2                                                                        +                                                            Im                  ⁢                                                            {                                              b                        1                                            }                                        2                                                  +                                  Im                  ⁢                                                            {                                              a                        1                                            }                                        2                                                                                                          (        3        )            
One of the drawbacks associated with the foregoing technique is that the response of boundary orientation angles sometimes appear erratic, typically occurring when individual angles from the real and imaginary components have opposite signs. An alternative boundary orientation angle definition, derived using the orthogonality condition of a1 and b1 components in the bedding coordinate system as defined for Deep Directional Resistivity (DDR) measurements [see reference 3], is given by:
                    ϕ        =                              1            2                    ⁢                      tan                          -              1                                ⁢                                                                      a                  1                                ⁢                                  b                  1                  *                                            +                                                b                  1                                ⁢                                  a                  1                  *                                                                                                      a                  1                                ⁢                                  a                  1                  *                                            -                                                b                  1                                ⁢                                  b                  1                  *                                                                                        (        4        )            
In terms of voltages, the angle is:
                    ϕ        =                              1            2                    ⁢                      tan                          -              1                                ⁢                                                                      V                  zx                                ⁢                                  V                  zy                  *                                            +                                                V                  zy                                ⁢                                  V                  zx                  *                                                                                                                                            V                    ⁢                                                                                  ⁢                    zx                                                                    2                            -                                                                  Vzy                                                  2                                                                        (        5        )            
The boundary orientation may then be used to compose symmetrized and anti-symmetrized measurements [see references 1, 2 and 3]. The symmetrized measurements exactly remove the anisotropy and dip effect for electromagnetic induction measurements and approximately remove these effects for LWD electromagnetic propagation measurements in transversely isotropic formations. The antisymmetrized measurements minimize the boundary effect and enhance the anisotropy and dip effect.
To characterize the orientation (azimuth) of crossed fractures, aligned with the borehole, there are two approaches available for tri-axial induction logging measurements using crossed dipole XY and transverse couplings XX and YY [see reference 6] and LWD directional electromagnetic propagation with rotating LWD tools using a ratio of transverse antennas [see reference 7] and the second harmonic ratio XX/YY [see reference 8]. These are so-called second harmonic measurements. The same angles are also indicators of boundary orientation in TI layered formations.